The transition probability \(P(x^{\prime }|x)\)  is one probability distribution for every point \(x\in \mathbf {X}_{N}\) . This endows the configuration space \(\mathbf {X}_{N}\)  with the structure of a statistical manifold (see e.g. [1]}[2]}). The benefit of this insight is that there is a natural metric associated to a statistical manifold, the information metric, which is unique (see e.g. [3]} and [4]}) up to an overall scale factor.
